Issue with Cloth and Hook Connection for Animation
Your problem stems from the cloth physics cache only baking to frame 250, in the blend file you sent I increased this limit and now it stretches with the hook object as intended. I also did some complimentary mesh cleanup as many of your objects have bloated geometry and overlapping vertices. In future also remember to reset some value of your physics sim in order to reset and rebake it if you find things aren't changing.
Link to fixed blend file.
https://drive.google.com/file/d/1wwrORCaKK0TmbGw5CIhmpXpMH3zHCKp8/view?usp=sharing